Vector Product in Mathematics
The vector product, also known as the cross product, is a fundamental operation in vector algebra that produces a vector perpendicular to both of the input vectors.
NoteThis concept is crucial in various fields of mathematics, physics, and engineering.
Definition and Formula
The vector product of two vectors $\mathbf{v}$ and $\mathbf{w}$ is defined as:
$$ \mathbf{v} \times \mathbf{w} = |\mathbf{v}||\mathbf{w}| \sin \theta \mathbf{n} $$
Where:
- $|\mathbf{v}|$ and $|\mathbf{w}|$ are the magnitudes of the vectors
- $\theta$ is the angle between $\mathbf{v}$ and $\mathbf{w}$
- $\mathbf{n}$ is the unit normal vector perpendicular to both $\mathbf{v}$ and $\mathbf{w}$
The direction of $\mathbf{n}$ is determined by the right-hand screw rule: if you curl the fingers of your right hand from $\mathbf{v}$ to $\mathbf{w}$ through the smaller angle, your thumb points in the direction of $\mathbf{n}$.
Properties of Vector Product
The vector product possesses several important properties:
- Anticommutativity: $\mathbf{v} \times \mathbf{w} = -(\mathbf{w} \times \mathbf{v})$ This means that changing the order of the vectors in a cross product negates the result.
- Distributivity over addition: $\mathbf{u} \times (\mathbf{v} + \mathbf{w}) = \mathbf{u} \times \mathbf{v} + \mathbf{u} \times \mathbf{w}$ This property allows us to break down complex cross products into simpler ones.
- Scalar multiplication: $(k\mathbf{v}) \times \mathbf{w} = k(\mathbf{v} \times \mathbf{w})$ Where $k$ is a scalar.
- Self-cross product: $\mathbf{v} \times \mathbf{v} = \mathbf{0}$ The cross product of a vector with itself is always the zero vector.
- Parallel vectors: For non-zero vectors, $\mathbf{v} \times \mathbf{w} = \mathbf{0}$ if and only if $\mathbf{v}$ and $\mathbf{w}$ are parallel.
